A positive and asymptotic preserving filtered PN method for the gray radiative transfer equations

Abstract

This paper presents a positive and asymptotic preserving scheme for the nonlinear gray radiative transfer equations. The scheme is constructed by combining the filtered spherical harmonics (FPN) method for the discretization of angular variable and with the framework of the unified gas kinetic scheme (UGKS) for the spatial- and time-discretization. The constructed scheme is almost free of ray effects and can also mitigate oscillations in the spherical harmonics (PN) approximation. Moreover, it can be shown that the current scheme is asymptotic preserving. Consequently, in the optically thick regimes the current scheme can exactly capture the solution of the diffusion limit equation without requiring the cell size being smaller than the photon's mean free path, while the solution in optically thin regimes can also be well resolved in a natural way. In addition, the FPN angular discretization induces a natural macro-micro decomposition, with this help we can obtain the sufficient conditions that guarantee the positivity of the radiative energy density and material temperature. Then, a linear scaling limiter is given to enforce that sufficient conditions. With the process of such construction, we finally obtain a scheme, called the PPFPN-based UGKS scheme, that is positive and asymptotic preserving. Various numerical experiments are included to validate the robustness, positive- and asymptotic-preserving property as well as the property of almost ray effect free.